Question

If arithmetic mean of two positive numbers is \[A\], their geometric mean is \[G\] and harmonic mean is \[H\], and then \[H\] is equal to
A. \[\frac{{{G^2}}}{A}\]
B. \[\frac{G}{{{A^2}}}\]
C. \[\frac{{{A^2}}}{G}\]
D. \[\frac{A}{{{G^2}}}\]

Answer 1

HINT:
We must use formulas for various types of means when answering questions of this nature. We are aware that the arithmetic mean (AM), geometric mean (GM), and harmonic mean are the three Pythagorean means (HM). Additionally, we are aware that the Arithmetic Mean (AM) equals \[\frac{{a + b}}{2}\], the Geometric Mean (GM) equals \[\sqrt {ab} \] and the Harmonic Mean (HM) equals \[\frac{{2ab}}{{(a + b)}}\] if a and b are two positive numbers.
Formula use:
if a and b are two positive numbers.
the Geometric Mean (GM) equals \[\sqrt {ab} \]
Arithmetic Mean (AM) equals \[\frac{{a + b}}{2}\]
Harmonic Mean (HM) equals \[\frac{{2ab}}{{(a + b)}}\]
Complete step-by-step solution
Now, we have to determine the relationship between the arithmetic, geometric, and harmonic means of two different positive real numbers denoted by \[A,G,H\] respectively.
Assuming that a and b are the two separate positive real numbers, we have
Arithmetic mean as,
\[ \Rightarrow AM = A = \frac{{(a + b)}}{2}\]
We have geometric mean as,
\[ \Rightarrow GM = G = \sqrt {ab} \]
We have Harmonic mean as,
\[ \Rightarrow HM = H = \frac{{2ab}}{{(a + b)}}\]
Thus, from the previous calculations, it is understood that,
\[ = \frac{{{G^2}}}{A}\]
We can prove that by substituting the values of \[G\] and \[A\] in the above formula, we get
\[ = \frac{{{{\left( {\sqrt {ab} } \right)}^2}}}{{\frac{{a + b}}{2}}}\]
Now, we have to apply the fraction rule \[\frac{a}{{\frac{b}{c}}} = \frac{{a \cdot c}}{b}\]:
\[ = \frac{{{{\left( {\sqrt {ab} } \right)}^2} \cdot 2}}{{a + b}}\]
On applying exponent rule \[{\left( {\sqrt {ab} } \right)^2} = ab\], we get
\[ = \frac{{ab \cdot 2}}{{a + b}}\]
Now, we have to rewrite the above expression, we get
\[ = \frac{{2ab}}{{a + b}}\]
Therefore, the value of \[H\] is equal to \[\frac{{{G^2}}}{A}\]
Hence, the option A is correct.
NOTE:
Students must be able to recall the arithmetic mean, geometric mean, and harmonic mean formulas for this type of question. When the distinction between GM and HM as well as between AM and GM is simplified, students must exercise caution. Students should also be aware that both differences are obviously greater than or equal to zero because they take the form of squares.